homework9 2010 solutions

# 021699 3 212 1039682 0006 0004911 3 221 1039682 00255

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0.021699 3 2,1,2 1.039682 0.006 0.004911 3 2,2,1 1.039682 0.0255 0.020871 3 2,2,2 1.092727 0.1445 0.107068 The incomes y t ( λ t ), probabilities π t ( λ t ), and contingent claims prices q t ( λ t ) are computed using the above formulae. The price of a riskless bond at time t is simply the sum of all contingent claims prices to consumption at t . See the column of the table labeled BOND. For comparison, note that β = 0.96, β 2 = 0.9216, and β 3 = 0.884736, so the rate of return on a bond is less than the discount rate. d) The price of a bond at time t contingent on λ t = λ 1 is obtained by summing over the prices of contingent claims for histories where λ t = λ 1 . See the column of the table labeled BOND 1. e) The price of a bond at time t contingent on λ t = λ 2 is obtained by summing over the prices of contingent claims for histories where λ t = λ 2 . See the column of the table labeled BOND 2. f) Clearly the price of a riskless bond is the sum of the price of a bond that pays off if λ t = λ 1 and the price of a bond that pays off if λ t = λ 2 since these are the only two possibilities. 2) An economy consists of two infinitely-lived consumers named i = 1, 2. There is one nonstorable consumption good. Consumer i consumes c i t at time t and ranks consumption streams by = 0 ) ( t t i t c u β , where β (0, 1) and u ( c ) is increasing, strictly concave, and C 2 . Consumer 1 is endowed with a stream of consumption y 1 t = (1, 0, 0, 1, 0, 0, 1, . . . ). Consumer 2 is endowed with a stream of consumption y 2 t = (0, 1, 1, 0, 1, 1, 0, . . . ). Assume there are complete markets with time 0 trading.

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ECN/APEC 7240 Spring 2010 a) A competitive equilibrium consists of an allocation { c 1 t , c 2 t } and a price function { q t 0 } such that (1) for i = 1, 2, given q t 0 , c i t solves the problem = 0 ) ( max t t i t c u β subject to = = 0 0 0 0 t t i t t t i t y q c q and (2) for each t 0 . 1 2 1 2 1 = + + t t t
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